Int. Adv. Algebra Geometry Solving a Trig. Function Review Name:

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1 Int. Adv. Algebra Geometry Solving a Trig. Function Review Name: Solving a trigonometric function for all solutions depends on the trigonometric ratio you are trying to solve. Consider these three equations and their corresponding graphs below. sin x = 1 2 cos x = 1 2 tan x = 1 2 For sine, cosine, and tangent you can easily see that an infinite number of solutions are possible. However, each function differs slightly in how the solution is found. Let s consider each one separately: First, sin x = 1 2 The functions f 1 = sin x andf 2 = 1 2 are graphed on the window [ 2π, 2π], [ 1, 1]. To find the first solution using sin 1 x: sin 1 (sin x) = sin x = π The second solution, is found by subtracting π from π 6 or5π. This is due to 6 the symmetry of the sine curve. The solutions are π above 0 and π below π. 6 6 These two solutions will then repeat every period of the function, or every 2π. The final result is then x = π + 2πn, n is an integer 6 x = 5π + 2πn, n is an integer 6

2 Second, cos x = 1 2 The functions f 1 = cos x andf 2 = 1 2 are graphed on the window [ 2π, 2π], [ 1, 1]. To find the first solution using cos 1 x: cos 1 (cos x) = cos x = π The second solution, is found by subtracting π from 2π or5π curve. The solutions are π above 0 and π below 2π.. This is due to the symmetry of the cosine These two solutions will then repeat every period of the function, or every 2π. The final result is then x = π + 2πn, n is an integer x = 5π + 2πn, n is an integer

3 Lastly, tan x = 1 2 The functions f 1 = tan x andf 2 = 1 2 are graphed on the window [ 2π, 2π], [ 1, 1]. ***Note that the period of the tangent function is NOT 2π, but π. On this window you see 4 cycles of the tangent function,not 2 that were standard on the cosine and sine functions. This will affect how we will write the infinite number of solutions, for tangent we will need to add multiples of π. To find the first solution using tan 1 x: tan 1 (tan x) = tan x.464 This is NOT a value that can be written in terms of π, the tangent ratio is not equal to ½ for any of the special right triangles. Also...this is the ONLY solution on one cycle...tangent does not repeat any of its values on a period cycle. The solutions will then repeat every period of the function, or every π. The final result is then x = πn, n is an integer Summary: When finding all the solutions to the sine function you will need to use sin 1 to find the first solution, then subtract your result from π. Then add 2πn where n is an integer to show all of the solutions for each additional period of the sine function. When finding all the solutions to the cosine function you will need to use cos 1 to find the first solution, then subtract your result from 2π. Then add 2πn where n is an integer to show all of the solutions for each additional period of the cosine function. When finding all the solutions to the tangent function you will need to use tan 1 to find the only solution on the period of tangent. Then add πn where n is an integer to show all of the solutions for each additional period of the tangent function. ***You will only get out exact values (in terms of π) when using ratios from the special right triangles. These ratios are 0, ½,, 2 2 2, 1 for sine and cosine, and additionally for tangent 0,,, 1

4 Practice Problems: Find all solutions for x. ***You will only get out exact values (in terms of π) when using ratios from the special right triangles. These ratios are 0, ½,, 2 2 2, 1 for sine and cosine, and additionally for tangent 0,,, 1 1. sin x = sin x + 4 = sin x. sin x 2 = cos x 2 = 2 5. cos x = cos x = 0 7. cos x + 1 = 2 cos x

5 ***You will only get out exact values (in terms of π) when using ratios from the special right triangles. These ratios are 0, ½,, 2 2 2, 1 for sine and cosine, and additionally for tangent 0,,, 1 8. tan x = 1 9. tan x = Give the domain and range of each of the following functions: a. s(x) = sin 1 x Domain: Range: b. c(x) = cos 1 x Domain: Range: c. t(x) = tan 1 x Domain: Range:

6 Use a graphing calculator using inverse trig AND a graph to solve the following questions. 11. On a particular day, the depth of water in feet at the entrance to a harbor is modeled by the function d(t) = 8 + sin 0.5t, where t is hours after 7 A.M. What are the minimum and maximum depths on this day? When do they first occur after 7:00 A.M.? Minimum Depth: Maximum Depth: When is the depth at 7 feet? List all of the times between 7:00 A.M. and 11:00 P.M. on that day? 12. On a particular day, the depth of water in feet at the entrance to a harbor is modeled by the function d(t) = 12 4 cos(.55t), where t is hours after 5 A.M. What are the minimum and maximum depths on this day? When do they first occur after 5:00 A.M.? Minimum Depth: Maximum Depth: When is the depth of water at 9feet? List all of the times between 5 A.M and 5 P.M. on that day?